Superuid signatures in a dissipative quantum point contact Meng-Zi Huang1Jerey Mohan1Anne-Maria Visuri2Philipp Fabritius1Mohsen Talebi1Simon Wili1Shun Uchino3Thierry Giamarchi4and Tilman Esslinger1

2025-05-02 0 0 1.23MB 17 页 10玖币
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Superfluid signatures in a dissipative quantum point contact
Meng-Zi Huang*,1, †Jeffrey Mohan*,1Anne-Maria Visuri,2Philipp Fabritius,1Mohsen
Talebi,1Simon Wili,1Shun Uchino,3Thierry Giamarchi,4and Tilman Esslinger1
1Institute for Quantum Electronics, ETH Z¨urich, 8093 Z¨urich, Switzerland
2Physikalisches Institut, University of Bonn, Nussallee 12, 53115 Bonn, Germany
3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
4Department of Quantum Matter Physics, University of Geneva,
24 quai Ernest-Ansermet, 1211 Geneva, Switzerland
(Dated: Tuesday 23rd May, 2023)
We measure superfluid transport of strongly interacting fermionic lithium atoms through a quan-
tum point contact with local, spin-dependent particle loss. We observe that the characteristic
non-Ohmic superfluid transport enabled by high-order multiple Andreev reflections transitions into
an excess Ohmic current as the dissipation strength exceeds the superfluid gap. We develop a model
with mean-field reservoirs connected via tunneling to a dissipative site. Our calculations in the
Keldysh formalism reproduce the observed nonequilibrium particle current, yet do not fully explain
the observed loss rate or spin current.
The interplay between coherent Hamiltonian dynam-
ics and incoherent, dissipative dynamics emerging from
coupling to the environment leads to rich phenomena
in open quantum systems [1–3], including the quantum
Zeno effect [4–9], emergent dynamics [10–15], and dissi-
pative phase transitions [16–21]. Moreover, an important
question is how many-body coherence competes with dis-
sipation by dephasing or particle loss. Directed transport
between two reservoirs offers an advantageous setup for
studying this competition since dissipation can be ap-
plied locally without perturbing the many-body states in
the reservoirs [22]. So far, studies on dissipation in solid-
state systems have focused on dephasing [23–25]. More
recently, quantum gases have become versatile platforms
to study interacting many-body physics and to engineer
novel forms of dissipation [2,4], though previous trans-
port experiments on dissipation have focused on weakly
interacting systems [26–28].
Engineered dissipation in strongly correlated fermionic
systems, while only starting to be explored theoretically
[20,29], opens interesting themes such as its compe-
tition with superfluidity where pairing and many-body
coherence are key. While the Josephson effect is an
archetype of superfluid transport, irreversible currents
between superfluids are highly nontrivial but less stud-
ied in cold-atom systems. A prime example is the ex-
cess current between two superconductors [30] or super-
fluids [31] through a high-transmission quantum point
contact (QPC) under a chemical potential bias where the
Josephson current is suppressed. Because of the super-
fluid gap ∆, direct quasiparticle transport is suppressed
when the chemical potential difference ∆µbetween the
reservoirs is smaller than 2∆ [illustrated in Fig. 1(d)]. In-
stead, this energy barrier can be overcome by cotunneling
*These authors contributed equally to this work
†mhuang@phys.ethz.ch
of many Cooper pairs npair ≥∆/∆µ, each providing an
energy 2∆µ[32,33]. This process is known as multi-
ple Andreev reflections (MAR) [34–37]. The robustness
of MAR to dissipation is an interesting open question,
especially for pair-breaking particle loss acting on only
one spin state, since the very existence of MAR relies on
many-body coherence between the spins.
In this work, we address this question by experimen-
tally and theoretically studying the influence of spin-
dependent particle loss on superfluid transport. We use a
strongly correlated Fermi gas—a superfluid with many-
body pairing—in a transport setup with two reservoirs
connected by a QPC and apply controllable local par-
ticle loss at the contact. We find that, surprisingly,
the superfluid behavior survives for dissipation strengths
larger than ∆—the energy scale responsible for the ob-
served current. This result is reproduced by a minimal
model that includes both superconductivity and dissipa-
tion written in the Keldysh formalism.
Experiment.—We prepare a degenerate Fermi gas of
6Li in a harmonic trap in a balanced mixture of the first-
and third-lowest hyperfine ground states, labeled ∣↓⟩and
∣↑⟩. The atomic cloud has typical total atom numbers
N=N↓+N↑=195(14)×103, temperatures T=100(2)nK,
and Fermi temperatures TF=h¯νtrap(3N)1/3/kB=
391(10)nK, where his the Planck constant, kBthe Boltz-
mann constant, and ¯νtrap =98(2)Hz the geometrical
mean of the harmonic trap frequencies. Using a pair
of repulsive, TEM01-like beams intersecting at the cen-
ter of the cloud, we optically define two half-harmonic
reservoirs connected by a quasi-1D channel with trans-
verse confinement frequencies νx=10(2)kHz and νz=
9.9(2)kHz, realizing a QPC illustrated in Fig. 1(a). We
apply a magnetic field of 689.7 G to address the spins’
Feshbach resonance, giving rise to a fermionic superfluid
in the densest parts of the cloud at the contacts to the
1D channel. An attractive Gaussian beam propagating
along zacts as a “gate” potential Vgwhich increases the
arXiv:2210.03371v2 [cond-mat.quant-gas] 22 May 2023
2
FIG. 1. (a) Two-terminal transport setup of a strongly inter-
acting Fermi gas with a dissipation beam resonant with ∣↓⟩at
the center of a 1D channel. A gate beam (dashed circle) ex-
erts an attractive potential Vgthat determines the superfluid
gap ∆ and the number of transport modes nm. (b) Theo-
retical model where the channel is modeled by a single, lossy
site tunnel coupled to BCS reservoirs. (c) Dissipation scheme
showing the relevant atomic energy levels. ∣↓⟩is optically
excited by the dissipation beam to ∣e⟩which decays predomi-
nantly to an auxiliary ground state ∣5⟩that quickly leaves the
system. (d) Illustration of MAR in a superconducting QPC:
transporting a quasiparticle requires energy to overcome the
gap 2∆, which is enabled by cotunneling of many pairs, each
providing a small energy 2(µL−µR). The quasiparticle and
constituents of the pairs experience dissipation, inhibiting this
process.
local chemical potential at the contacts. It enhances the
local degeneracy T/TF∼0.03 [38] well into the superfluid
phase and determines both the superfluid gap ∆ and the
number nmof occupied transverse transport modes in
the contact. Both ∆ and nmcan be computed from the
known potential energy landscape and equation of state
[38] and are approximately ∆ ≈kB×1.4µK≈̵
h×184 ms−1
and nm≈3 in this work.
We engineer spin-dependent particle loss with a tightly
focused beam at the center of the 1D channel [Fig. 1(a)]
that optically pumps ∣↓⟩to an auxiliary ground state ∣5⟩
[Fig. 1(c)] which interacts weakly with the two spin states
and is lost due to photon recoil [38]. This leads to a
controllable particle dissipation rate of ∣↓⟩atoms given
by the peak photon scattering rate Γ↓with no observable
heating in the reservoirs. While this loss beam is far off-
resonant for ∣↑⟩, causing no dissipation in the absence
of interatomic interactions, we observe loss of ∣↑⟩in the
strongly-interacting regime studied here [38]. Even at
the strongest dissipation, the system lifetime is over a
second, much longer than the timescale of a detectable
transport of 103atoms via MAR ∼103h/∆∼30 ms.
We induce particle transport from the left to the
right reservoir by preparing an atom number imbalance
∆N=NL−NRthat generates a chemical potential bias
∆µ=∆µ(∆N, N, T )given by the system’s equation of
state [38]. ∆µdrives a current IN=−˙
∆N/2 from left to
right that causes ∆Nto decay over time t. The dynam-
ics of ∆N(t)/N(t)for various Γ↓are plotted in Fig. 2(a).
From each trace, we numerically extract the current-bias
relation IN(∆µ), shown in Fig. 2(b) in units of the su-
perfluid gap ∆ [38].
For weakly interacting spins, the decay of ∆N(t)at
arbitrary Γ↓is exponential since a QPC coupling two
Fermi liquids has a linear (Ohmic) current-bias relation
IN=G∆µ. The conductance Gis quantized in units of
2/h(2 comes from spin) but can be renormalized to a
smaller value by dissipation [27,58]. In contrast, we ob-
serve that the decay of ∆Nat Γ↓=0 deviates strongly
from an exponential and the corresponding current-bias
relation IN(∆µ)is highly nonlinear, consistent with pre-
vious observations in the strongly interacting regime [31].
In fact, this nonlinearity is a signature of superfluidity.
Specifically, for ballistic superconducting QPCs [36], the
current is approximately IN≈G∆µ+Iexc
Nwhere the nor-
mal current G∆µ≈2nm∆µ/his carried by quasiparti-
cles. The excess current Iexc
N≈(16/3)nm∆/hcarried by
the Cooper pairs is given by the superfluid gap ∆—the
natural energy scale in this system. The dominance of
the excess current over the normal current can be seen
in ∆N(t)/Nwith Γ↓=0: It decays almost linearly in
time, indicating that the current is nearly independent
of ∆µ. We do not expect Josephson oscillations (as in
Ref. [50,52]) here since the irreversible current (MAR)
damps the reversible Josephson current below our exper-
imental resolution [51,54].
We now turn to the question of the influence of dis-
sipation on the superfluid transport. At low Γ↓, super-
fluidity is still evident from the linear initial decay of
∆Nwhich gives way to exponential behavior at low bias
where the MAR responsible for transport become higher
order. The effect of dissipation is clearer in the extracted
current-bias relations shown in Fig. 2(b). As dissipa-
tion strength increases, the initial current (largest ∆µ)
reduces as does the concavity of the curve. The current-
bias relation eventually approaches a straight line at high
dissipation, yet the slope is still significantly larger than
the normal conductance 2nm/h[dashed line in Fig. 2(b)].
We argue below, after fitting our theoretical model to the
data, that this excess conductance, albeit linear, is a sig-
nature of the MAR process. An increased temperature
leading to a suppressed gap can result in similar anoma-
lous conductance [31,73,76–78], though we can exclude
this effect here from the absence of temperature increase
in the reservoirs. Another possible contribution to the
linear conductance in a unitary Fermi gas arises from
pair tunneling coupled to collective modes in the super-
fluid [51,56]. However, it is expected to be on the order
of 2/hper mode, thus negligible compared to the MAR
contribution even at strong dissipation. This is also con-
sistent with the small linear conductance (slope at large
bias ∼2nm/h) in the observed current-bias relation at
Γ↓=0.
Theoretical model and fit.—We have developed a mean-
field model based on previous work [31] by adding an
atom loss process within the Lindblad master equation
3
FIG. 2. (a) Time evolution of particle imbalance for various
dissipation strengths Γ↓. Solid curves are fits of the theoret-
ical model and the data sets are vertically offset by 0.1 for
clarity. Error bars represent 1σstatistical uncertainty of 4
to 5 repetitions. The fitted dissipation γσ(γ↑≡0.72γ↓) for
each curve is plotted versus Γ↓in the inset, showing a linear
relation (solid lines are linear fits). (b) Numerically extracted
current-bias relation for the same data sets in units of the gap
∆≈̵
h×184 ms−1. The dashed line indicates the maximum nor-
mal conductance 2nm/h. Error bars include uncertainties in
the conversion to reduced units. The uncertainties in the cal-
culated current, represented by the shaded bands, originate
mostly from the number of transmission modes nm.
framework [38]. The two reservoirs are treated as BCS
superfluids, and transport through the channel is mod-
eled by single-particle tunneling with amplitude τfrom
either reservoir onto a dissipative site between them
[Fig. 1(b)]. The dissipation is modeled with a Lindblad
operator ˆ
Lσ=√γσˆ
dσproportional to the site’s fermionic
annihilation operator ˆ
dσwith a rate γσfor each spin
σ=↓,↑, i.e. as a pure particle loss process that is uncor-
related between the two spins. γ↓is directly related to
Γ↓as our results below show, while γ↑is included phe-
nomenologically to match the experimental observation
that ∣↑⟩is also lost due to the strong interaction.
To compute nonequilibrium observables such as IN,
we use the Keldysh formalism extended to dissipative
systems [20,41–45,79]. The time integral of the the-
oretical current IK
N(∆µ, τ, γ↓, γ↑)along with the equa-
tion of state ∆µ(∆N, N, T )yields the model’s predic-
tion for the time evolution of the particle imbalance
∆NK(t, τ, γ↓, γ↑). Here, N(t)is obtained from an ex-
ponential fit to the data [38]. For simplicity, we model
a single transport mode and obtain the net current by
multiplying by the number of modes ˙
∆NK=−2nmIK
N.
Although the formalism can treat nonzero temperatures,
we simplify the calculation by using zero temperature
since kBT/∆<0.08.
Because of the spin-dependent loss, a spin imbalance
builds up, leading to a magnetization imbalance ∆M=
∆N↓−∆N↑≠0 that can drive additional particle current.
Nevertheless, ∆Mremains small (∆M/N<0.07) for all
our data and its effect on INis negligible [38]. Moreover,
the model predicts a spin current 1 order of magnitude
below the observed value, and therefore does not fully
describe the spin-dependent transport. In this work, we
focus on spin-averaged particle transport.
We perform a least-squares fit of ∆NK(t, τ, γ↓, γ↑)to
the measured ∆N(t)to extract the model parameters τ,
γ↓, and γ↑. In our case of low bias and ballistic channel,
IK
Nis sensitive only to the sum γ=γ↑+γ↓. Therefore,
to avoid overfitting, we fix the ratio γ↑/γ↓=rto the av-
erage measured ratio of the atom loss rates of the two
spin states r=˙
N↑/˙
N↓=0.72(4)[38]. As the gap ∆ is
the natural unit of current and bias, the estimation of
∆ in the experimental system is crucial for quantitative
comparison to the model. However, the spatially varying
gap in the potential energy landscape of the experiment
(crossover from the 3D reservoirs to the 1D channel) is
not explicitly modeled. Motivated by this and an exper-
imental uncertainty of about 6% in the potential energy
of the gate beam, we fit a multiplicative correction factor
ηgon the gate potential Vg, which strongly affects both
our estimate of ∆ in the most degenerate point in the
system and nm[38]. Since the system is identical in each
dataset except for the dissipation strength, we first fit
the Γ↓=0 data with γ↓=γ↑=0 to find τand ηg, which
determine the concavity of IK
N(∆µ)and the timescale of
∆NK(t), respectively. We then fit the single parameter
γ↓with fixed r,τ, and ηgfor subsequent sets. The sys-
tematic error in ηgis the dominant source of uncertainty
in our theoretical calculations.
The fits for each dataset are plotted as solid lines
in Figs. 2(a),(b). With this fitting procedure, we find
ηg=0.98(6), ∆/kB=1.4(1)µK, and nm=2.6(5). The
fitted τdetermines the energy linewidth of the dissi-
pative site Γd∝τ2[38], which in units of the gap is
Γd=5.9(4)∆. Independent measurements with different
values of νxand Vgproduce similar results for ηgand
Γdwithin 20%. The large value of Γdreflects the near-
perfect transmission of the ballistic channel as the limit
Γd→∞is equivalent to perfect transparency α→1 in a
QPC model with direct tunneling between the reservoirs
[31,38,40]. In principle, the energy dof the dissipa-
tive site is another free parameter. Because of the large
linewidth, however, the model is insensitive to changes of
dwithin the physically meaningful range ∣d∣<∆. There
are thus no resonant effects as in a weakly coupled quan-
4
tum dot, and we fix d=0. For the same reason, we do
not consider any on-site interaction.
The fitted γ↓and γ↑versus Γ↓, plotted in the inset of
Fig. 2(a), show that the effective dissipation strength is
approximately proportional to the photon scattering rate
as expected. The fitted slope γ↓=3.6Γ↓is of order 1,
corroborating the use of our single-site model where only
one ∣↓⟩fermion fits into the dissipation beam at a time. In
accordance, the dissipation beam’s waist wy=1.31(2)µm
is comparable to the Fermi wavelength in the channel
λF≈2µm.
Robustness of superfluid transport to dissipation.—
Supported by the overall fit to the data, our calculations
provide insight into the observed flattening of the non-
linear current-bias relation: a plausible scenario is that
dissipation suppresses higher-order MAR processes while
allowing lower-order MAR (npair <∆/∆µ) to contribute
[38]. Despite the nonlinearity disappearing as dissipa-
tion increases, the current still originates from MAR. A
benchmark for this superfluid signature is the excess cur-
rent above the possible normal current 2nm∆µ/h. To see
this quantitatively, we replot the measured and calcu-
lated current in Fig. 3(a) versus Γ↓at a few bias values
including the initial bias ∆µ/∆≈0.05 (npair ≈20) down
to ∆µ/∆≈0.005 (npair ≈200). The fitted model is shown
in solid curves using the fitted linear relationship between
γ↓and Γ↓in the inset of Fig. 2(a). The observed current
well exceeds the upper bound of normal current (dashed
lines in corresponding colors) for all biases, indicating
the persistence of the MAR-enabled current. Moreover,
the excess current decays smoothly with dissipation and
gives no indication of a dissipative phase transition but
rather shows a dissipative superfluid-to-normal crossover
(cf. Ref.[6,25]).
Since the current IN=−(˙
NL−˙
NR)/2 could in principle
arise purely from an asymmetric particle loss, we verify
that the conserved current Icons—the atoms transported
through the QPC without being lost—exceeds the upper
bound for the normal current at dissipation strengths
above the gap. This is shown in Fig. 3(b) where we plot
the data for the largest bias together with bounds for
Icons. These bounds are obtained by writing ˙
NL/R=
∓Icons −Iloss
L/R, where Iloss
L/Ris the dissipation-induced cur-
rent into the vacuum [45], and thus IN=Icons +(Iloss
L−
Iloss
R)/2. Assuming the worst-case scenario—maximally
asymmetric loss—leads to Icons ∈[IN−˙
N/2, IN+˙
N/2].
The excess conserved current shows that the system pre-
serves its superfluid signature up to ̵
hγ ≳∆.
Atom loss rate.—Finally, we compare the theoretical
model to the experiment in terms of the total atom loss
rate ˙
N=˙
NL+˙
NR. In the Keldysh formalism, comput-
ing steady-state observables involves an integration over
energy. Because our model assumes reservoirs with lin-
ear dispersion, we introduce a high energy cutoff Λ. The
current only has contributions from energies constrained
by ∆µand ∆, and is therefore independent of the cut-
off when Λ >∆,∆µ. The atom loss rate on the other
FIG. 3. (a) Measured current vs. dissipation strength at dif-
ferent biases. The theoretical calculations are shown as solid
curves with uncertainties in lighter colors. Dashed lines rep-
resent upper bounds of normal-state currents 2nm∆µ/h. (b)
Same data at ∆µ/∆≈0.05 with bounds on the conserved cur-
rent Icons ∈[IN−˙
N/2, IN+˙
N/2]indicated by the shaded area
(lighter color represents the uncertainty). The horizontal axis
is the fitted γ=γ↓+γ↑in units of the gap to illustrate the
effective strength of the dissipation. The conserved current is
above the normal current (bounds represented by the horizon-
tal bar), showing superfluid character even for the strongest
dissipation.
hand depends on the cutoff and converges to its maxi-
mum value when Λ ≫Γd. We find that a single Λ cannot
reproduce the observed atom loss rate at different dissi-
pation [shown in Fig. 4(a) for ∆µ/∆≈0.05]: The atom
loss rate saturates as dissipation increases, while the cal-
culated loss rate increases almost linearly with dissipa-
tion in the measured range. In particular, Λ ≳20∆ is
needed to reproduce the data at weak dissipation, close
to the converged value, while Λ ≈5∆ reproduces the
data at strong dissipation. This discrepancy is shown
in Fig 4(b) which plots the Λ needed to reproduce ˙
N
at each measured Γ↓. Physically, this means that addi-
tional mechanisms not included in the model suppress
the occupation of the dissipative site and hence the loss
rate. The calculations do predict a saturation of ˙
Nat an
order of magnitude higher dissipation strength followed
by a slow decrease of ˙
Nversus Γ↓—a signature of the
quantum Zeno effect [4–6]. We do not observe this non-
monotonicity, and expect it to be washed out due to the
lack of energy resolution in ˙
N[80] and the finite width
of the dissipative beam.
Conclusion.—We have shown a remarkable robust-
ness of the MAR processes responsible for the current
through a QPC between two superfluids of strongly in-
teracting Fermi gas under spin-dependent particle loss.
We observed no critical behavior, which contrasts with
other types of superfluid-suppressing perturbations such
as moving defects [81–83], magnetic fields in supercon-
ductors [84], and photon absorption by superconducting
nanowire single-photon detectors [85]. While our most
significant observations are captured by our model, devi-
ations from the theory point to the importance of strong
摘要：

SuperuidsignaturesinadissipativequantumpointcontactMeng-ZiHuang*,1,„Je reyMohan*,1Anne-MariaVisuri,2PhilippFabritius,1MohsenTalebi,1SimonWili,1ShunUchino,3ThierryGiamarchi,4andTilmanEsslinger11InstituteforQuantumElectronics,ETHZurich,8093Zurich,Switzerland2PhysikalischesInstitut,UniversityofBonn,N...

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